To implement fault-tolerant quantum computation (FTQC) with continuous variables, continuous variables need to be digitized using an appropriate code such as the Gottesman-Kitaev-Preskill (GKP) qubit. The scheme introduced in [Fukui et al. Phys. Rev. X 8, 021054 (2018)] has reduced the threshold of the squeezing level required for continuous-variable FTQC to less than 10 dB, assuming noise derived from the GKP qubit itself. In this paper, we propose a scheme to improve noise tolerance during the construction of a large-scale cluster state used for FTQC with the GKP qubits. In our scheme, a small-scale cluster state is prepared by employing maximum-likelihood estimation, the entanglement generation via the Bell measurement, and probabilistic reliable measurement. Then, a large-scale cluster state is constructed from the small-scale cluster states via the reliable encoded Bell measurement. In the numerical calculations, we assume errors derived from the two-mode gate and loss in the homodyne measurement in addition to noise from the GKP qubit itself. The results show that the thresholds of a squeezing level are around 8.1, 9.6, and 12.4 dB for loss in the homodyne measurement 0, 5, and 10%, respectively. Hence, this paper provides a way toward continuous-variable FTQC with a feasible squeezing level.
To implement fault-tolerant quantum computation (FTQC) with continuous variables, continuous variables need to be digitized using an appropriate code such as the Gottesman-Kitaev-Preskill (GKP) qubit. The scheme introduced in [Fukui et al. Phys. Rev. X 8, 021054 (2018)] has reduced the threshold of the squeezing level required for continuous-variable FTQC to less than 10 dB, assuming noise derived from the GKP qubit itself. In this paper, we propose a scheme to improve noise tolerance during the construction of a large-scale cluster state used for FTQC with the GKP qubits. In our scheme, a small-scale cluster state is prepared by employing maximum-likelihood estimation, the entanglement generation via the Bell measurement, and probabilistic reliable measurement. Then, a large-scale cluster state is constructed from the small-scale cluster states via the reliable encoded Bell measurement. In the numerical calculations, we assume errors derived from the two-mode gate and loss in the homodyne measurement in addition to noise from the GKP qubit itself. The results show that the thresholds of a squeezing level are around 8.1, 9.6, and 12.4 dB for loss in the homodyne measurement 0, 5, and 10%, respectively. Hence, this paper provides a way toward continuous-variable FTQC with a feasible squeezing level.
Quantum computation with light, compared with other platforms, offers the unique benefit of natural high-speed operations at room temperature and large clock rate, but a big obstacle of photonics is the lack of strong nonlinearities which also makes loss-tolerant or generally fault-tolerant quantum computation (FTQC) complicated in an all-optical setup. Typical current approaches to optical FTQC that aim at building suitable large multi-qubit cluster states by linearly fusing small elementary resource states would still demand either fairly expensive initial resources or rather low loss and error rates. Here we propose reintroducing weakly nonlinear operations, such as a weak cross-Kerr interaction, to achieve small initial resource cost and high error thresholds at the same time. More specifically, we propose an approach to generate a large-scale cluster state by hybridizing Gottesman-Kitaev-Preskill (GKP) and single-photon qubits. Our approach enables us to implement FTQC based on GKP squeezing of 7.4 and 8.4 dB and a photon loss rate of 1.0 and 5.0 %, respectively. In addition, our scheme has a reduced resource cost, i.e., number of physical qubits/photons per logical qubit or initial entanglement, compared to high-threshold FTQC with optical GKP qubits or fusion-based quantum computation with encoded single-photon-qubit states, respectively. Furthermore, our approach, when assuming very low photon loss, allows to employ GKP squeezing as little as 3.8 dB, which cannot be achieved by using GKP qubits alone.
Hybridizing different physical systems or degrees of freedom offers significant advantages for realizing practical, universal, scalable, and fault-tolerant quantum computation (FTQC). Here, we propose optical FTQC schemes with low squeezing thresholds by leveraging the strengths of both discrete-variable (DV) and continuous-variable (CV) systems while utilizing frequency, time, and orbital angular momentum degrees of freedom. First, we design an optical entanglement generator (OEG) capable of producing various types of entangled pairs, including cluster pairs, hybrid entangled pairs, and GKP Bell pairs, which can be flexibly chosen by adjusting the measurement basis. Additionally, the OEG features extra ports for directly inputting (outputting) data (result) states via quantum teleportation, eliminating the need for optical switches. Secondly, large-scale one-dimensional, two-dimensional, and three-dimensional (3D) hybrid cluster states, composed of DV Gottesman-Kitaev-Preskill (GKP) qubits and CV squeezed states, are deterministically generated using the entangled pairs passed through a time-delay system. Moreover, we optimize the surface-GKP code to further reduce logical errors during the stabilizer measurements in the surface code. By combining the 3D cubic hybrid cluster state with the modified surface-GKP code and accounting for full circuit-level noise, FTQC is achieved with a squeezing threshold of 10 dB. Moreover, our method can also generate a 3D macronode Raussendorf-Harrington-Goyal (RHG) cluster state, facilitating an alternative FTQC scheme via the RHG-GKP code. Our work provides a viable pathway toward future optical FTQC architectures.
Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction.However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction.In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation.To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources.Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation.To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state.We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.
Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction. However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction. In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation. To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources. Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation. To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state. We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.Received 11 January 2021Revised 18 May 2021Accepted 23 July 2021DOI:https://doi.org/10.1103/PRXQuantum.2.030325Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasMeasurement-based quantum computingOptical quantum information processingQuantum information processing with continuous variablesQuantum InformationAtomic, Molecular & Optical
In order to solve practical problems on a quantum computer, it is necessary to use fault-tolerant quantum error correction schemes to overcome noise: the errors arising from imperfections in physical components. The Gottesman-Kitaev-Preskill (GKP) code aims at achieving this in a hardware efficient manner by encoding finite-dimensional logical subspaces in the Hilbert space of one or more continuous variable modes. In near term implementations, however, it is not feasible to eliminate errors entirely, so it is natural to also employ alternative error mitigation techniques together with error correction. In this work, we study a quantum error mitigation method known as probabilistic error cancellation in the context of the GKP code. We compare Steane-type and teleportation-based GKP error correction, and calculate the sampling overheads associated with the technique for square and hexagonal GKP codes. We employ the stabilizer subsystem decomposition for GKP codes to obtain an effective logical channel for noisy operations that is used to express the ideal one of a target logical unitary. We consider noise from finite squeezing on the data and the two ancilla modes, needed for error correction, and examine the relationship between the sampling overhead and the noise for different decoding methods. Our results are calculated for single- and two-qubit GKP Clifford gates and one round of error correction, and show the nontrivial combination of error correction and mitigation in continuous variable codes.
The Gottesman-Kitaev-Preskill (GKP) qubit is a promising ingredient for fault-tolerant quantum computation (FTQC) in optical continuous variables due to its advantage of noise tolerance and scalability. However, one of the main problems in the preparation of the optical GKP qubit is the difficulty in obtaining the nonlinearity. Cross-Kerr interaction is one of the promising candidates for this nonlinearity. There is no existing scheme to use the cross-Kerr interaction to generate the optical GKP qubit for FTQC. In this work, we propose a generation method of the GKP qubit by using a cross-Kerr interaction between a squeezed light and a superposition of Fock states. We numerically show that the GKP qubit with the 10 dB can be generated with mean fidelities of 99.99 and 99.9% at the success probabilities of 2.7 and 4.8%, respectively. Therefore, our method has the potential to generate the optical GKP qubit with a quality required for FTQC when we obtain the sufficient technologies for the preparation of ancillary Fock states and a cross-Kerr interaction.